system and method for estimating uncertainty by obtaining conditional probability densities of bathymetric uncertainty due to navigation error and bottom slope using a monte carlo technique, using the monte carlo results to train a bayesian network to provide the causal relationships of navigation error and bottom slope to bathymetric uncertainty, producing a histogram of bathymetric uncertainty from the bayesian network of the uncertainty for an area similar to the training set area, and estimating the uncertainty based on the histogram produced by the bayesian network.
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9. A computer software system comprising computer code stored on non-transitory computer readable media for estimating uncertainty in an electronic bathymetry database, said computer code comprising:
obtaining conditional probability densities of bathymetric uncertainty due to navigation error and bottom slope using a monte carlo technique on representative sets of soundings data from the bathymetry database, the representative sets being associated with a training set area;
training a bayesian network to provide the causal relationships of the navigation error and the bottom slope to bathymetric uncertainty based on the conditional probability densities;
producing a histogram of the bathymetric uncertainty from the bayesian network for an area similar to the training set area; and
estimating an uncertainty based on the histogram.
1. A computer-implemented method for estimating uncertainty in a bathymetry database comprising:
obtaining, by a special-purpose computer, conditional probability densities of bathymetric uncertainty due to horizontal error and bottom slope using a monte carlo technique on representative sets of soundings data from the bathymetry database, the representative sets being associated with a training set area;
training, by the special-purpose computer, a bayesian network to provide causal relationships between the horizontal error and the bottom slope and the bathymetric uncertainty based on the conditional probability densities;
creating, by the special-purpose computer, a first histogram of the bathymetric uncertainty from the bayesian network for an area similar to the training set area; and
estimating, by the special-purpose computer, an uncertainty based on the first histogram.
6. An automated system for estimating uncertainty in an electronic bathymetry database comprising:
an automated conditional probability density processor obtaining conditional probability densities of bathymetric uncertainty due to navigation error and bottom slope using a monte carlo technique on representative sets of soundings data from the bathymetry database, the representative sets being associated with a training set area;
an automated bayesian network trainer processor training a bayesian network to provide the causal relationships of the navigation error and the bottom slope to the bathymetric uncertainty based on the conditional probability densities, said trainer processor producing a histogram of the bathymetric uncertainty from the bayesian network for an area similar to the training set area; and
an automated uncertainty estimator estimating an uncertainty based on the output histogram.
12. A computer system for providing uncertainty estimates to a bathymetry database comprising:
a bathymetry database having navigation error data and bathymetry data for a region of interest;
a bayesian network trainer processor obtaining track lines in the region of interest based on the navigation data, calculating fractional lengths of the track lines with like errors, computing weights for a horizontal error node based on the sums of the track lengths with the like errors divided by the total of the track lengths, extracting bathymetry data from the region of interest to obtain the magnitude of the bottom gradient, assigning the bottom gradient to a bin of a bottom gradient node, and providing the horizontal error node and the bottom gradient node to the bayesian network, the bayesian network providing a histogram based on the horizontal error node and the bottom gradient node; and
an uncertainty estimator calculating an intermediate uncertainty based on a statistic of the histogram, calculating a final uncertainty estimate based on adding the square of the intermediate uncertainty to the square of a vertical error estimate and computing either the square root of the sum or the square root multiplied by a pre-selected magnification coefficient to account for higher uncertainty in purely interpolated values, and populating a bathymetry database with the bathymetry data and the final uncertainty estimate.
14. A computer software system comprising computer code stored on non-transitory computer readable media for providing uncertainty estimates to a bathymetry database, said computer code comprising:
defining a region of interest for an uncertainty estimate;
accessing navigation data and bathymetry data for the region of interest;
obtaining track lines in the region of interest based on the navigation data;
calculating fractional lengths of the track lines with like errors;
computing weights for a horizontal error node based on the sums of the track lengths with the like errors divided by the total of the track lengths;
extracting bathymetry data from the region of interest to obtain the magnitude of the bottom gradient;
assigning the bottom gradient to a bin of a bottom gradient node;
providing the horizontal error node and the bottom gradient node to the bayesian network;
extracting a histogram from the bayesian network based on the horizontal error node and the bottom gradient node;
calculating an intermediate uncertainty based on a statistic of the histogram;
calculating a final uncertainty estimate based on adding the square of the intermediate uncertainty to the square of a vertical error estimate and computing either the square root of the sum or the square root multiplied by a pre-selected magnification coefficient to account for higher uncertainty in purely interpolated values; and
populating a bathymetry database with the bathymetry data and the final uncertainty estimate.
4. An automated method for providing uncertainty estimates to a bathymetry database comprising:
defining, by a special-purpose computer, a region of interest for an uncertainty estimate;
accessing, by the special-purpose computer, navigation data and bathymetry data for the region of interest;
obtaining, by the special-purpose computer, track lines in the region of interest based on the navigation data;
calculating, by the special-purpose computer, fractional lengths of the track lines with like errors;
computing, by the special-purpose computer, weights for a horizontal error node based on the sums of the track lengths with the like errors divided by the total of the track lengths;
extracting, by the special-purpose computer, bathymetry data from the region of interest to obtain the magnitude of the bottom gradient;
assigning, by the special-purpose computer, the bottom gradient to a bin of a bottom gradient node;
providing, by the special-purpose computer, the horizontal error node and the bottom gradient node to the bayesian network;
extracting, by the special-purpose computer, a histogram from the bayesian network based on the horizontal error node and the bottom gradient node;
calculating, by the special-purpose computer, an intermediate uncertainty based on a statistic of the histogram;
calculating, by the special-purpose computer, a final uncertainty estimate based on adding the square of the intermediate uncertainty to the square of a vertical error estimate and
computing either the square root of the sum or the square root multiplied by a pre-selected magnification coefficient to account for higher uncertainty in purely interpolated values; and
populating, by the special-purpose computer, a bathymetry database with the bathymetry data and the final uncertainty estimate.
2. The method as in
3. The method as in
(a) perturbing, by the special-purpose computer, positions of the sounding data a pre-selected number of times;
(b) obtaining, by the special-purpose computer, standard deviations of the pre-selected number of bathymetry layers from the perturbed positions;
(c) increasing, by the special-purpose computer, the horizontal error by a predefined step size and repeating steps (a) and (b); and
(d) creating, by the special-purpose computer, a conditional probabilities table of the standard deviations.
5. The method as in
7. The system as in
8. The system as in
10. The computer system as in
11. The computer system as in
(a) perturbing positions of the sounding data a pre-selected number of times;
(b) obtaining standard deviations of the pre-selected number of bathymetry layers from the perturbed positions;
(c) increasing the horizontal error by a predefined step size and repeating steps (a) and (b); and
(d) creating a conditional probabilities table of the standard deviations.
13. The computer system as in
15. The computer software system as in
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This application claims the benefit of priority based on U.S. Provisional Patent Application No. 61/333,795 filed on May 12, 2010, the entirety of which is hereby incorporated by reference into the present application.
Devices and methods disclosed herein relate generally to data accuracy, and more specifically, to computing uncertainty for gridded data sets, for example, for historical gridded bathymetry data.
Estimates of uncertainty are becoming a requirement of oceanographic and acoustic models that use bathymetry. Further, bathymetry fusion algorithms that fuse disparate data sets into a single bathymetry surface can require uncertainty estimates of the input data. Still further, International Hydrographic Organization (IHO) standards prescribe that uncertainty be specified for all hydrographic and bathymetric products, with differing level of uncertainty tolerances depending on safety requirements. Ultimately, uncertainty in the bathymetry layer can be used for navigation safety for surface ships and submarine operations. Jakobsson et al., On the effect of random errors in gridded bathymetric compilations, Journal of Geophysical Research-Solid Earth, 107: Article 2358, 2002, estimate error on historic data sets based on Monte Carlo simulations where the two-dimensional position of the original data points, the soundings, are randomly perturbed using a normally distributed random number generator (RNG) illustrated in
What are needed are a system and method that can estimate uncertainty in the interpolation/extrapolation of bathymetry data. What are further needed are a system and method that provide a statistically rigorous means for estimation of uncertainty for areas of the seafloor not covered by dedicated surveys or that fall in between point measurement locations, and that are computationally efficient and do not require the use of original soundings data.
To address the above-stated needs, the present teachings provide a system and method for estimating uncertainty based on a Bayesian network (BN). According to Heckerman, A Tutorial on Learning with Bayesian Networks, MICROSOFT® Technical Report, MSR-TR-95-06, March 1995 (revised November 1996), “a Bayesian network is a graphical model that encodes probabilistic relationships among variables of interest.” The BN can accommodate missing data and can learn causal relationships, thus it can include probabilistic semantics. A BN can encode uncertain expert knowledge in expert systems.
The present teachings can provide a computationally efficient method for estimating bathymetric uncertainty for historical gridded bathymetry data sets. Uncertainty estimates are needed when data are stored in the Bathymetric Attributed Grid files, which require both bathymetry and uncertainty. An exemplary embodiment of the present teachings, the Digital Bathymetry Data Base, Variable Resolution Uncertainty Expert System (DUES), is based on a BN to provide a computationally efficient method for determining uncertainty in, for example, but not limited to, the Navy's Digital Bathymetric Data Base-Variable Resolution (DBDB-V). In the present teachings, the Monte Carlo technique can be used on representative sets of soundings data to obtain the conditional probability densities (CPDs) necessary for statistical inference. Causal relationships of navigation error and bottom slope to bathymetric uncertainty can be quantified by CPD's.
The computer-based system for estimating uncertainty can include, but is not limited to including, an automated conditional probability density processor computing conditional probability densities of bathymetric uncertainty due to navigation error and bottom slope using a Monte Carlo technique on representative sets of soundings data from the bathymetry database. The system can also include an automated BN trainer processor using the Monte Carlo results to train the BN to provide the causal relationships of navigation error and bottom slope to bathymetric uncertainty, producing a histogram of bathymetric uncertainty from the Bayesian Network of the uncertainty for an area similar to the training set area, and an automated uncertainty estimator estimating the uncertainty based on the histogram produced by the Bayesian Network, providing the uncertainty estimates to an upgraded bathymetry database.
The method for estimating uncertainty can include, but is not limited to including, the steps of obtaining conditional probability densities of bathymetric uncertainty due to navigation error and bottom slope using a Monte Carlo technique on representative sets of soundings data from the bathymetry database, using the Monte Carlo results to train the BN to provide the causal relationships of navigation error and bottom slope to bathymetric uncertainty, producing a histogram of bathymetric uncertainty from the Bayesian Network of the uncertainty for an area with similar bottom topography to the training set area, and estimating the uncertainty based on the histogram produced by the Bayesian Network. Similarity is quantified, for example, but not limited to, by statistical hypothesis testing of the distributions of the bottom slopes in one area versus the training area such that the null hypothesis cannot be rejected due to lack of evidence for rejection at a 99% percentile confidence level. The expert system of the present teachings is fundamentally different from established Monte Carlo procedure because statistical inference is used to estimate uncertainty whereas Monte Carlo uses standard deviation from simulations, and while Monte Carlo simulations can be used for training, Monte Carlo simulation is not the means by which the uncertainties are estimated. Further, original soundings are not required to estimate the uncertainty.
The problems set forth above as well as further and other problems are solved by the present teachings. These solutions and other advantages are achieved by the various embodiments of the teachings described herein below.
In the present embodiment, an established Monte Carlo technique can be used on representative sets of soundings data to obtain the CPD's necessary for the statistical inference. The BN can then produce a histogram of this uncertainty estimate for an area given the navigation errors used to survey the region and bottoms slopes that are present. A final estimate of uncertainty can be calculated by combining a variance estimate, such as, for example, but not limited to, mean plus one standard deviation or a quantile, from the BN's output histogram with the vertical error, VK, under the assumption of statistical independence between the two. The statistic can be user-supplied.
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In the present embodiment, N=100 Monte Carlo iterations are performed for each mth set of simulations; the set of one hundred Monte Carlo simulations are then repeated M times for each horizontal error category in
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The present embodiment is directed, in part, to software for accomplishing the methods discussed herein, and computer readable media storing software for accomplishing these methods. The various modules described herein can be accomplished on the same CPU, or can be accomplished on different computers. In compliance with the statute, the present embodiment has been described in language more or less specific as to structural and methodical features. It is to be understood, however, that the present embodiment is not limited to the specific features shown and described, since the means herein disclosed comprise preferred forms of putting the present embodiment into effect.
Referring again primarily to
Although the present teachings have been described with respect to various embodiments, it should be realized these teachings are also capable of a wide variety of further and other embodiments.
Elmore, Paul A., Fabre, David H., Sawyer, Raymond T., Ladner, Rodney W.
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